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AQA Further Paper 2 2020 June Q3
1 marks Moderate -0.5
Find the gradient of the tangent to the curve $$y = \sin^{-1} x$$ at the point where \(x = \frac{1}{5}\) Circle your answer. [1 mark] \(\frac{5\sqrt{6}}{12}\) \quad \(\frac{2\sqrt{6}}{5}\) \quad \(\frac{4\sqrt{3}}{25}\) \quad \(\frac{25}{24}\)
AQA Further Paper 2 2020 June Q4
3 marks Standard +0.8
The matrices \(\mathbf{A}\) and \(\mathbf{B}\) are defined as follows: $$\mathbf{A} = \begin{bmatrix} x + 1 & 2 \\ x + 2 & -3 \end{bmatrix}$$ $$\mathbf{B} = \begin{bmatrix} x - 4 & x - 2 \\ 0 & -2 \end{bmatrix}$$ Show that there is a value of \(x\) for which \(\mathbf{AB} = k\mathbf{I}\), where \(\mathbf{I}\) is the \(2 \times 2\) identity matrix and \(k\) is an integer to be found. [3 marks]
AQA Further Paper 2 2020 June Q5
5 marks Standard +0.3
Solve the inequality $$\frac{2x + 3}{x - 1} \leq x + 5$$ [5 marks]
AQA Further Paper 2 2020 June Q6
5 marks Challenging +1.2
Find the sum of all the integers from 1 to 999 inclusive that are not square or cube numbers. [5 marks]
AQA Further Paper 2 2020 June Q7
5 marks Standard +0.8
The diagram shows part of the graph of \(y = \cos^{-1} x\) \includegraphics{figure_7} The finite region enclosed by the graph of \(y = \cos^{-1} x\), the \(y\)-axis, the \(x\)-axis and the line \(x = 0.8\) is rotated by \(2\pi\) radians about the \(x\)-axis. Use Simpson's rule with five ordinates to estimate the volume of the solid formed. Give your answer to four decimal places. [5 marks]
AQA Further Paper 2 2020 June Q8
9 marks Hard +2.3
  1. Factorise \(\begin{vmatrix} 2u + h + x & x + h & x^2 + h^2 \\ 0 & a & -a^2 \\ a + b & b & b^2 \end{vmatrix}\) as fully as possible. [6 marks]
  2. The matrix \(\mathbf{M}\) is defined by $$\mathbf{M} = \begin{bmatrix} 13 + x & x + 3 & x^2 + 9 \\ 0 & 5 & 25 \\ 8 & 3 & 9 \end{bmatrix}$$ Under the transformation represented by \(\mathbf{M}\), a solid of volume \(0.625 \text{m}^3\) becomes a solid of volume \(300 \text{m}^3\) Use your answer to part (a) to find the possible values of \(x\). [3 marks]
AQA Further Paper 2 2020 June Q9
7 marks Challenging +1.8
The matrix \(\mathbf{C} = \begin{bmatrix} a & -b \\ b & a \end{bmatrix}\), where \(a\) and \(b\) are positive real numbers, and \(\mathbf{C}^2 = \begin{bmatrix} \frac{\sqrt{3}}{2} & -\frac{1}{2} \\ \frac{1}{2} & \frac{\sqrt{3}}{2} \end{bmatrix}\) Use \(\mathbf{C}\) to show that \(\cos \frac{\pi}{12}\) can be written in the form \(\frac{\sqrt{m + n}}{2}\), where \(m\) and \(n\) are integers. [7 marks]
AQA Further Paper 2 2020 June Q10
6 marks Challenging +1.2
The sequence \(u_1, u_2, u_3, \ldots\) is defined by $$u_1 = 0 \quad u_{n+1} = \frac{5}{6 - u_n}$$ Prove by induction that, for all integers \(n \geq 1\), $$u_n = \frac{5^n - 5}{5^n - 1}$$ [6 marks]
AQA Further Paper 2 2020 June Q11
8 marks Challenging +1.2
  1. Starting from the series given in the formulae booklet, show that the general term of the Maclaurin series for $$\frac{\sin x}{x} - \cos x$$ is $$(-1)^{r+1} \frac{2r}{(2r + 1)!} x^{2r}$$ [4 marks]
  2. Show that $$\lim_{x \to 0} \left[ \frac{\sin x}{x} - \cos x \right] \frac{1}{1 - \cos x} = \frac{2}{3}$$ [4 marks]
AQA Further Paper 2 2020 June Q12
12 marks Challenging +1.3
  1. Given that \(I = \int_a^b e^{2t} \sin t \, dt\), show that $$I = \left[ qe^{2t} \sin t + re^{2t} \cos t \right]_a^b$$ where \(q\) and \(r\) are rational numbers to be found. [6 marks]
  2. A small object is initially at rest. The subsequent motion of the object is modelled by the differential equation $$\frac{dv}{dt} + v = 5e^t \sin t$$ where \(v\) is the velocity at time \(t\). Find the speed of the object when \(t = 2\pi\), giving your answer in exact form. [6 marks]
AQA Further Paper 2 2020 June Q13
10 marks Challenging +1.2
Charlotte is trying to solve this mathematical problem: Find the general solution of the differential equation $$\frac{d^2y}{dx^2} + \frac{dy}{dx} - 2y = 10e^{-2x}$$ Charlotte's solution starts as follows: Particular integral: \(y = \lambda e^{-2x}\) so $$\frac{dy}{dx} = -2\lambda e^{-2x}$$ and $$\frac{d^2y}{dx^2} = 4\lambda e^{-2x}$$
  1. Show that Charlotte's method will fail to find a particular integral for the differential equation. [2 marks]
  2. Explain how Charlotte should have started her solution differently and find the general solution of the differential equation. [8 marks]
AQA Further Paper 2 2020 June Q14
11 marks Hard +2.3
The diagram shows the polar curve \(C_1\) with equation \(r = 2 \sin \theta\) The diagram also shows part of the polar curve \(C_2\) with equation \(r = 1 + \cos 2\theta\) \includegraphics{figure_14}
  1. On the diagram above, complete the sketch of \(C_2\) [2 marks]
  2. Show that the area of the region shaded in the diagram is equal to $$k\pi + m\alpha - \sin 2\alpha + q \sin 4\alpha$$ where \(\alpha = \sin^{-1} \left( \frac{\sqrt{5} - 1}{2} \right)\), and \(k\), \(m\) and \(q\) are rational numbers. [9 marks]
AQA Further Paper 2 2020 June Q15
16 marks Challenging +1.2
The points \(A(7, 2, 8)\), \(B(7, -4, 0)\) and \(C(3, 3.2, 9.6)\) all lie in the plane \(\Pi\).
  1. Find a Cartesian equation of the plane \(\Pi\). [3 marks]
  2. The line \(L_1\) has equation \(\mathbf{r} = \begin{bmatrix} 5 \\ -0.4 \\ 4.8 \end{bmatrix} + \mu \begin{bmatrix} 15 \\ 3 \\ 4 \end{bmatrix}\)
    1. Show that \(L_1\) lies in the plane \(\Pi\). [2 marks]
    2. Show that every point on \(L_1\) is equidistant from \(B\) and \(C\). [4 marks]
  3. The line \(L_2\) lies in the plane \(\Pi\), and every point on \(L_2\) is equidistant from \(A\) and \(B\). Find an equation of the line \(L_2\) [4 marks]
  4. The points \(A\), \(B\) and \(C\) all lie on a circle \(G\). The point \(D\) is the centre of circle \(G\). Find the coordinates of \(D\). [3 marks]
AQA Further Paper 2 2023 June Q1
1 marks Easy -1.8
Given that \(y = \sin x + \sinh x\), find \(\frac{d^2y}{dx^2} + y\) Circle your answer. [1 mark] \(2\sin x\) \quad \(-2\sin x\) \quad \(2\sinh x\) \quad \(-2\sinh x\)
AQA Further Paper 2 2023 June Q2
1 marks Easy -1.8
Which one of the expressions below is not equal to zero? Circle your answer. [1 mark] \(\lim_{x \to \infty} (x^2e^{-x})\) \quad \(\lim_{x \to 0} (x^5 \ln x)\) \quad \(\lim_{x \to \infty} \left(\frac{e^x}{x^5}\right)\) \quad \(\lim_{x \to 0^+} (x^3e^x)\)
AQA Further Paper 2 2023 June Q3
1 marks Standard +0.3
The determinant \(A = \begin{vmatrix} 1 & 1 & 1 \\ 2 & 0 & 2 \\ 3 & 2 & 1 \end{vmatrix}\) Which one of the determinants below has a value which is not equal to the value of \(A\)? Tick (\(\checkmark\)) one box. [1 mark] \(\begin{vmatrix} 3 & 1 & 3 \\ 2 & 0 & 2 \\ 3 & 2 & 1 \end{vmatrix}\) \quad \(\square\) \(\begin{vmatrix} 1 & 2 & 3 \\ 1 & 0 & 2 \\ 1 & 2 & 1 \end{vmatrix}\) \quad \(\square\) \(\begin{vmatrix} 2 & 2 & 2 \\ 1 & 0 & 1 \\ 3 & 2 & 1 \end{vmatrix}\) \quad \(\square\) \(\begin{vmatrix} 1 & 1 & 1 \\ 3 & 2 & 1 \\ 2 & 0 & 2 \end{vmatrix}\) \quad \(\square\)
AQA Further Paper 2 2023 June Q4
1 marks Easy -1.2
It is given that \(f(x) = \cosh^{-1}(x - 3)\) Which of the sets listed below is the greatest possible domain of the function \(f\)? Circle your answer. [1 mark] \(\{x : x \geq 4\}\) \quad \(\{x : x \geq 3\}\) \quad \(\{x : x \geq 1\}\) \quad \(\{x : x \geq 0\}\)
AQA Further Paper 2 2023 June Q5
5 marks Challenging +1.2
Josh and Zoe are solving the following mathematics problem: The curve \(C_1\) has equation $$\frac{x^2}{16} - \frac{y^2}{9} = 1$$ The matrix \(\mathbf{M} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\) maps \(C_1\) onto \(C_2\) Find the equations of the asymptotes of \(C_2\) Josh says that to solve this problem you must first carry out the transformation on \(C_1\) to find \(C_2\), and then find the asymptotes of \(C_2\) Zoe says that you will get the same answer if you first find the asymptotes of \(C_1\), and then carry out the transformation on these asymptotes to obtain the asymptotes of \(C_2\) Show that Zoe is correct. [5 marks]
AQA Further Paper 2 2023 June Q6
5 marks Standard +0.3
  1. Express \(-5 - 5\text{i}\) in the form \(re^{i\theta}\), where \(-\pi < \theta \leq \pi\) [2 marks]
  2. The point on an Argand diagram that represents \(-5 - 5\text{i}\) is one of the vertices of an equilateral triangle whose centre is at the origin. Find the complex numbers represented by the other two vertices of the triangle. Give your answers in the form \(re^{i\theta}\), where \(-\pi < \theta \leq \pi\) [3 marks]
AQA Further Paper 2 2023 June Q7
3 marks Standard +0.8
Show that $$\sum_{r=11}^{n+1} r^3 = \frac{1}{4}(n^2 + an + b)(n^2 + an + c)$$ where \(a\), \(b\) and \(c\) are integers to be found. [3 marks]
AQA Further Paper 2 2023 June Q8
6 marks Standard +0.3
\(\mathbf{A}\) is a non-singular \(2 \times 2\) matrix and \(\mathbf{A}^T\) is the transpose of \(\mathbf{A}\)
  1. Using the result $$(\mathbf{AB})^T = \mathbf{B}^T\mathbf{A}^T$$ show that $$(\mathbf{A}^{-1})^T = (\mathbf{A}^T)^{-1}$$ [3 marks]
  2. It is given that \(\mathbf{A} = \begin{pmatrix} 4 & 5 \\ -1 & k \end{pmatrix}\), where \(k\) is a real constant.
    1. Find \((\mathbf{A}^{-1})^T\), giving your answer in terms of \(k\) [2 marks]
    2. State the restriction on the possible values of \(k\) [1 mark]
AQA Further Paper 2 2023 June Q9
7 marks Challenging +1.2
The complex number \(z\) is such that $$z = \frac{1 + \text{i}}{1 - k\text{i}}$$ where \(k\) is a real number.
  1. Find the real part of \(z\) and the imaginary part of \(z\), giving your answers in terms of \(k\) [2 marks]
  2. In the case where \(k = \sqrt{3}\), use part (a) to show that $$\cos \frac{7\pi}{12} = \frac{\sqrt{2} - \sqrt{6}}{4}$$ [5 marks]
AQA Further Paper 2 2023 June Q10
8 marks Challenging +1.2
The region \(R\) on an Argand diagram satisfies both \(|z + 2\text{i}| \leq 3\) and \(-\frac{\pi}{6} \leq \arg(z) \leq \frac{\pi}{2}\)
  1. Sketch \(R\) on the Argand diagram below. [3 marks] \includegraphics{figure_10a}
  2. Find the maximum value of \(|z|\) in the region \(R\), giving your answer in exact form. [5 marks]
AQA Further Paper 2 2023 June Q11
9 marks Standard +0.8
The line \(l_1\) passes through the points \(A(6, 2, 7)\) and \(B(4, -3, 7)\)
  1. Find a Cartesian equation of \(l_1\) [2 marks]
  2. The line \(l_2\) has vector equation \(\mathbf{r} = \begin{pmatrix} 8 \\ 9 \\ c \end{pmatrix} + \mu \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}\) where \(c\) is a constant.
    1. Explain how you know that the lines \(l_1\) and \(l_2\) are not perpendicular. [2 marks]
    2. The lines \(l_1\) and \(l_2\) both lie in the same plane. Find the value of \(c\) [5 marks]
AQA Further Paper 2 2023 June Q12
6 marks Standard +0.3
The function \(f\) is defined by $$f(n) = 3^{3n+1} + 2^{3n+4} \quad (n \in \mathbb{Z}^+)$$ Prove by induction that \(f(n)\) is divisible by 19 for \(n \geq 1\) [6 marks]